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Chapter 2: Q53E (page 83)

A certain shop repairs both audio and video components. Let A denote the event that the next component brought in for repair is an audio component, and let B be the event that the next component is a compact disc player (so the event B is contained in A). Suppose that \(P\left( A \right) = 0.6\)and\(P\left( B \right) = 0.05\). What is\(P\left( {B|A} \right)\)?

Short Answer

Expert verified

The probability, \(P\left( {A|B} \right) = 0.0833\)

Step by step solution

01

Definition of Probability

The term "probability" simply refers to the likelihood of something occurring. We can talk about the probabilities of certain outcomes鈥攈ow likely they are鈥攚hen we're unsure about the outcome of an event. Statistics is the study of occurrences guided by probability.

02

Calculation for the determination of probability

In the exercise, we are given that

\(\begin{array}{l}P(A) &=& 0.6\\P(B) &=& 0.05\\B \subseteq A\end{array}\)

From \(B \subseteq A\)

we have

\(A \cap B = B \Rightarrow P(A \cap B) = P(B) = 0.05\)

Conditional probability of A given that the event B has occurred, for which \(P(B) > 0\),\(P(A\mid B) = \frac{{P(A \cap B)}}{{P(B)}}\)\(\)

for any two events A and B.

From the definition we have

\(P(B\mid A) = \frac{{P(B \cap A)}}{{P(A)}} = \frac{{P(B)}}{{P(A)}} = \frac{{0.05}}{{0.6}} = 0.0833\)

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Most popular questions from this chapter

Reconsider the system defect situation described in Exercise.

a. Given that the system has a type \(1\) defect, what is the probability that it has a type \({\bf{2}}\) defect?

b. Given that the system has a type \(1\) defect, what is the probability that it has all three types of defects?

c. Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect?

d. Given that the system has both of the first two types of defects, what is the probability that it does not have the third type of defect?

Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a \({\rm{CD}}\) player. Let \({{\rm{A}}_{\rm{1}}}\) be the event that the receiver functions properly throughout the warranty period, \({{\rm{A}}_{\rm{2}}}\) be the event that the speakers function properly throughout the warranty period, and \({{\rm{A}}_{\rm{3}}}\) be the event that the \({\rm{CD}}\) player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with \({\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ = }}{\rm{.95}}\), \({\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.98}}\), and \({\rm{P}}\left( {{{\rm{A}}_{\rm{3}}}} \right){\rm{ = }}{\rm{.80}}\).

a. What is the probability that all three components function properly throughout the warranty period?

b. What is the probability that at least one component needs service during the warranty period?

c. What is the probability that all three components need service during the warranty period?

d. What is the probability that only the receiver needs service during the warranty period?

e. What is the probability that exactly one of the three components needs service during the warranty period?

f. What is the probability that all three components function properly throughout the warranty period but that at least one fails within a month after the warranty expires?

The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group-blood group combinations.

Suppose that an individual is randomly selected from the population, and define events by \({\rm{A = }}\)\{type A selected), \({\rm{B = }}\) (type B selected), and \({\rm{C = }}\) (ethnic group \({\rm{3}}\) selected).

a. Calculate\({\rm{P(A),P(C)}}\), and \({\rm{P(A{C}C)}}\).

b. Calculate both \({\rm{P(A}}\mid {\rm{C)}}\)and \({\rm{P(C}}\mid {\rm{A)}}\), and explain in context what each of these probabilities represents.

c. If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group\({\rm{1}}\)?

A quality control inspector is examining newly produced items for faults. The inspector searches an item for faults in a series of independent fixations, each of a fixed duration. Given that a flaw is actually present, let p denote the probability that the flaw is detected during any one fixation (this model is discussed in 鈥淗uman Performance in Sampling Inspection,鈥� Human Factors, \({\rm{1979: 99--105)}}{\rm{.}}\)

a. Assuming that an item has a flaw, what is the probability that it is detected by the end of the second fixation (once a flaw has been detected, the sequence of fixations terminates)?

b. Give an expression for the probability that a flaw will be detected by the end of the nth fixation.

c. If when a flaw has not been detected in three fixations, the item is passed, what is the probability that a flawed item will pass inspection?

d. Suppose \({\rm{10\% }}\) of all items contain a flaw (P(randomly chosen item is flawed) . \({\rm{1}}\)). With the assumption of part (c), what is the probability that a randomly chosen item will pass inspection (it will automatically pass if it is not flawed, but could also pass if it is flawed)?

e. Given that an item has passed inspection (no flaws in three fixations), what is the probability that it is actually flawed? Calculate for \({\rm{p = 5}}\).

A box in a supply room contains \({\rm{15}}\) compact fluorescent lightbulbs, of which \({\rm{5}}\) are rated \({\rm{13}}\)-watt, \({\rm{6}}\)are rated \({\rm{18}}\)-watt, and \({\rm{4}}\) are rated \({\rm{23}}\)-watt. Suppose that three of these bulbs are randomly selected.

a. What is the probability that exactly two of the selected bulbs are rated \({\rm{23}}\)-watt?

b. What is the probability that all three of the bulbs have the same rating?

c. What is the probability that one bulb of each type is selected?

d. If bulbs are selected one by one until a \({\rm{23}}\)-watt bulb is obtained, what is the probability that it is necessary to examine at least 6 bulbs?
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